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Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation.

Bessel’s eq.

The solutions of the Bessel’s eq. are called Bessel functions.

In Chapter 3, we get the series solution of the above eq.

6.1 Bessel Functions of the First Kind,

* Generating function, integer order, Jn (x)

Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function. Let us introduce a function of two variables,

(6.1)

Expanding it in a Laurent series, we obtain

(6.2)

The coefficient of t n, Jn, is defined to be Bessel function of the first kind of integer order n. Expanding the exponentials, we have

(6.3)

Setting n =r- s, yields

(6.4)

Since

for a positive integer m)

(note:

The coefficient of t n is then

(6.5)

This series form exhibits the behavior of the Bessel function Jn for small x. The results for J0, J1, and J2are shown in Fig.6.1. The Bessel functions oscillate but are not periodic.

.

Figure 6.1 Bessel function,

, and

Eq.(6.5) actually holds for n < 0 , also giving

(6.6)

Since the terms for s<n (corresponding to the negative integer (s-n) ) vanish, the series can be considered to start with s=n. Replacing s by s + n, we obtain

(6.7)

These series expressions may be used with n replaced by v to define Jv and J-v for non-integer v.

* Recurrence relations

Differentiating Eq.(6.1) partially with respect to t, we findthat

(6.9)

and substituting Eq.(6.2) for the exponential and equating the coefficients of tn-1, we obtain

(6.10)

This is a three-term recurrence relation.

On the other hand, differentiating Eq.(6.1) partially with respect to x, we have

(6.11)

Again, substituting in Eq.(6.2) and equating the coefficients of t n , we obtain the result

(6.12)

As a special case,

(6.13)

Adding Eqs.(6.10) and (6.12) and dividing by 2, we have

(6.14)

Multiplying by xn and rearranging terms produces

(6.15)

Subtracting Eq.(6.12) from (6.10) and dividing by 2 yields

(6.16)

Multiplying by x-n and rearranging terms, we obtain

(6.17)

*Bessel’s differential equation

Please verify the follow result in class.

In particular, we have shown that the functions Jn defined by our generating functions, satisfy Bessel’s eq., and thus are indeed Bessel functions

A particular useful and powerful way of treating Bessel functions employs integral representations. If we return to the generating function (Eq. (6.2)), and substitute t=eiθ,

(6.23)

in which we have used the relations

(6.24)

and so on.

(6.25)

equating real and imaginary parts, respectively. It might be noted that angleθ (in radius) has no dimensions. Likewise sinθ has no dimensions and function cos(xsinθ) is perfectly proper from a dimensional point of view.

By employing the orthogonality properties of cousine and sine,

(6.26a)

(6.26b)

in which n and m are positive integers (zero is excluded), weobtain

repeats itself in all four quadrants (

,

), we may write Eq. (6.30) as

(6.30a)

On the other hand,

reverses its sign in the third and fourth quadrants

so that

Adding Eq. (6.30a) and i times Eq. (6.30b), we obtain the complex exponential representation

(6.30b)

(6.30c)

This integral representation (Eq. (6.30c)) may be obtained somewhat more directly by employing contour integration.

•Example 6.11 Fraunhofer Diffraction, Circular Aperture

In the theory of diffraction through a circular aperture we encounter the integral

(6.31)

for

, the amplitude of the diffracted wave. Here

is an azimuth angle in the

plane of the circular aperture of radius a,and

, is the angle defined by a

point on ascreen below the circular aperture relative to the normal through the center point. The parameter b is given by

(6.32)

with

defined by

the wavelength of the incident wave. The other symbols are

Fig.6.2 From Eq. (6.30c), we get

(6.33)

Equation (6.15) enables us to integrate Eq. (6.33) immediately toobtain

(6.34)

The intensity of the light in the diffraction pattern is proportional to

and

(6.35)

6.2 Orthogonality

For v> 0, Jv (0)=0. Thus, for a finite interval [0, a], when

zero of Jv(i.e.

), we are able to have

if m≠n,

(6.49)

This gives us orthogonality over the interval [0, a].

* Normalization

The normalization result may be written as

(6.50)

* Bessel series

If we assume that the set of Bessel functions

(v fixed, m=1,2,… )

may be expanded in a

is complete, then any well-behaved function

Bessel series

(6.51)

The coefficients cvm are determined by using Eq.(6.50),

(6.52)

* Continuum form

If a→∞, then the series forms may be expected to go over into integrals. The discrete roots become a continuous variable . A key relation is the Bessel function closure equation

(6.59)

6.3 Neumann function, Bessel function of the second kind,

From the theory of the differential equations it is known that Bessel’s equation has two independent solutions, Indeed, for non-integral order v we have already found two solutions and labeled themand ,using

theinfinite series (Eq. (6.5)). The trouble is that when v is integral Eq.(6.8) holds and we have but one independent solution. A second solution may be developed by the method of Section 3.6. This yields a perfectly good solution of Bessel’s equation but is not the usual standard form.

Definition and series form

As an alternate approach, we have the particular linear combination of \'

and

(6.60)

This is Neumann function (Fig. 6.3). For nonintegralv ,

clearly satisfies

Bessel’s equation, for it is a linear combination of known solutions,

and

, our Neumann function or Bessel function of the second

To verify that

kind, actually does satisfy Bessel’s equation for integral n , we mayprocess

as follows. L’Hospital’s rule applied to Eq. (6.60) yields

(6.65)

Differentiating Bessel’f equation for

with repect to v , we have

(6.66)

Multiplying the equation for

by(-1)v , substracting from the equation

(as suggested by Eq. (6.65)), and taking the limit

, we obtain

for

(6.67)

For

, an integer, the right-hand side vanishes by Eq. (6.8) and

is seen to be a solution of Bessel’s equation. The most general solution for

any v can be written as

(6.68)

Example Coaxial Wave Guides

We are interested in an electromagnetic wave confined between the conducting cylindrical surfaces

concentric ,

. Most of the mathematics is

and

inSection 3.3. From EM knowledge,

worked out

(: electrical field along z axis)

Let, we have

This is the Bessel equation. If ,the solution is

with

. But, for the coaxial wave guide one generalization

is needed. Theorigin is now excluded (). Hence the

Neumann functionmay not be excluded.

becomes

(6.79)

With the condition

(6.80)

we have the basic equatios for a TM (transverse magnetic ) wave.

The (tangential) electric field must vanish at the conducting surfaces(Direchlet

boundary condition) or

(6.81)

(6.82)

these transcendental equations may be solved for

and the ratio .

From the relation

(6.83)

and since must be positive for a real wave, the minimum

frequency that will

be propagated (in this TM mode) is

(6.84)

(6.82). This is

with fixed by the boundary conditions, Eqs. (6.81) and

the cutoff frequency of the wave guide.

Many authors perfer to introduce the Hankel functions by means of integral representations and then use them to define the Neumann function, .

We here introduce them a simple way as follows.

As we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions,

and

:

(6.85)

(6.86)

This is exactly analogous to taking

(6.87)

For real arguments and are complex conjugates. The extent of the analogy will be seen better when the asymptotic forms are considered . Indeed, it is their asymptotic behavior that makes the Hankel functuions useful!

6.5 Modified Bessel function , and

The Helmholtz equation,

separated in circular cylindrical coordinates, leads to Eq. (6.22a), the Bessel

equation. Equation (6.22a) is satisfied by the Bessel and Neumann functions

and

and any linear combination such as the Hankelfunctions

and

.Now the Helmholtz equation describes the space

part of wave phenomena. If instesd we have a diffusion problem, then the Helmholtz equation is replaced by

. (6.88)

(6.89)

The Helmholtz equation may be transformed into the diffusion equation by the transformation . Similarly,changes Eq. (6.22a) into Eq. (6.89) and shows that

The solution of Eq. (6.89) are Bessel function of imaginary argument. To obtain a solution that is regular at the origin, we takeas the regular Bessel function

. It is customary (and convenient) to choose the normalization sothat

(6.90)

(Here the variable is being replaced by x for simplicity.) Often this is written as

(6.91)

In the terms of infinite series this is equivalent to removing the

sign in Eq. (6.5) and writing

(6.92)

The extra

normalization cancels the

from each term and leaves

real. For integral v this yields

(6.93)

Recurrence relations

The recurrence relations satisfied by

may be developed from the series

expansions,but it is easier to work from the existing recurrence relations for

. Let us replace x by –ix and rewrite Eq. (6.90) as

(6.94)

Repalcing x by ix, we have a recurrence relation for ,

Equation (6.12) transforms to

(6.95)

(6.96)

From Eq. (6.93) it is seen that we have but one independent solution when v is an integer, exactly as in the Bessel function solution of Eq. (6.108) is essentially a matter od convenience.

We choose to define a second solution in terms of the Hankel function

by

(6.97)

The factor makesreal when x is real. Using Eqs. (6.60) and (6.90), we may transform Eq. (6.97) to

(6.98)

analogous to Eq. (6.60) for The choice of Eq. (6.97) as a definition is somewhat unfortunate in that the function does not satisfy the same recurrence relations as . To avoid this annoyance other authors have included an additional factor of cos. This permits satisfy the same recurrence relations as , but it has the disadvantage of making for

.

To put the modified Bessel functions and in proper perspective, we introduce them here because:

1. These functions are solutions of the frequently encountered modified Bessel equation.

2. They are needed for specific physical problems such as diffusion problems.

Figure 6.4Modified Bessel functions

6.6 Asmptotic behaviors

Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions (see page 450 in the text book for details) and related functions:

1.

2. The second kind Hankel function is just the complex conjugate of the

first (for real argument),

(6.100)

3. Since is the real part of

(6.101)

4. The Neumann function is the imaginary partof

, or

(6.102)

5. Finally, the regular hyperbolic or modified Bessel function

is given by

(6.103)

or

(6.104)

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